Authorea

Comparative statics

We investigate here some positive properties of the equilibrium of the contest.

Using (LABEL:eq:x*) and (LABEL:eq:y*), we can calculate the aggregate lobbying effort, the probability of approving the new product and the ex post expected social surplus¹¹Note that we implicitly assume here that the expected social surplus is the sum of the expected utilities of players $I$ and $E$, (\ref{eq:uI}) and (\ref{eq:uE}) respectively. This is justified if the new product has a perfect substitute, which price is exogenously given, or if it is sold on the foreign market., respectively equal to

\begin{equation} x^{\ast}\left(\delta\right)+y^{*}=\begin{cases}\frac{\left(1-k\right)^{2}\left(b-k\right)}{2\left(b-k\right)+\left(1-k\right)^{2}}\sqrt{\frac{b-\delta}{b-k}},&\mbox{if $\delta\leq k,$}\\ \frac{\left(1-k\right)^{2}\left(b-k\right)}{2\left(b-k\right)+\left(1-k\right)^{2}},&\mbox{otherwise}.\end{cases}\nonumber \\ \end{equation} \begin{equation} \pi\left(x^{\ast}\left(\delta\right),y^{*}\right)=\begin{cases}1-\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}}\sqrt{\frac{b-k}{b-\delta}},&\mbox{if $\delta\leq k,$}\\ 1-\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}},&\mbox{otherwise}.\end{cases}\nonumber \\ \end{equation} \begin{equation} \left(\begin{array}{c}\pi\left(x^{\ast}\left(\delta\right),y^{*}\right)\left(b-\delta\right)\\ -x^{*}\left(\delta\right)-y^{*}\end{array}\right)=\begin{cases}\left(b-\delta\right)-2\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}}\sqrt{\left(b-\delta\right)\left(b-k\right)},&\mbox{if $\delta\leq k,$}\\ \left(b-\delta\right)-\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}}\left(2b-\delta-k\right),&\mbox{otherwise}.\end{cases}\nonumber \\ \end{equation}

Let $\phi\left(b,k\right)$ be defined by $\phi\left(b,k\right)\equiv 8\left(b-k\right)^{2}+\left(1-k\right)\left(1-2b+k^{2}\right)$. The following property characterizes the comparative statics with respect to $k$. The proof is given in the Appendix.

Property 1. (Comparative statics). Assume that $b$ and $k$ are such that $\phi\left(b,k\right)>0$. The comparative statics with respect to $k$ satisfies:

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The aggregate lobbying effort is decreasing in $k$;
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The probability of approving the product is increasing in $k$ if and only if $\delta\leq k$ or if $\delta>k$ and $b>\left(1+k\right)/2$;
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The ex post expected social surplus is increasing in $k$.